It means that p-value is less than alpha, where alpha is an acceptable upper limit to the magnitude of type I error. In the criminal justice context, the type I error is a conviction of an innocent person.

I think that if you want to measure it in probabilities, we need to know how often such “events” happen. If it is in, say, data transmissions, and the probabilities you list is for the probability that a bit is flipped, then none of your options are “beyond reasonable doubt”.

This is of course a question of values, but to answer it one has to estimate the tradeoff between the false positive and false negative rates. I suspect that these are very different for different crimes. For instance in ‘he said she said’ kind of crimes (like the one our beloved former president commited) I suspect that if you insist on a confidence of less than a few percent you will never convict anyone and people would be able to commit these crimes with immunity. On the other hand crimes that typically leave physical evidence (like say financial crimes) I would have liked a confidence of 0.001 or lower.

I think it should be obvious that p-value is irrelevant due to Lindley’s Paradox. Relying on p-values can lead to the conclusion that someone who is guilty with a probability of one in a million is guilty beyond reasonable doubt.
By the way, Gil, your title (and apparently the possible answers you have in mind) is about “what does *beyond* reasonable doubt mean “, whereas your question asks about “reasonable doubt”, which I guess is the complimentary.

I don’t see Lindley’s Paradox as necessarily implying that p-value is irrelevant. One can interpret this paradox as saying that the Bayesian approach is problematic due to its sensitivity to the choice of H1, the alternative hypothesis (i.e., the choice of the prior). However, there is always the issue of base rates: a person having the exact DNA sequence that was found in the crime scene and that is shared by only 1/1,000,000 of the population does not have a 1/1,000,000 probability to be innocent, especially if his name was drawn in search for a DNA match from a computer archive containing the DNA sequences of the 10M residents of the city…

One interesting comment I heard was about people who were asked to answer certain yes/no questions and to indicate how certain they are about the answers. 100% certainty was a fairly common answer. Then they were further asked to estimate in how many questions where they gave 100% certainty their answer was correct. The estimates were around 80%.

Sure. We should actually be before we say such things. The standards of justice should be greater than scrutiny of medicine, because some people have the view when people with good intentions and finite resources have been allowed to revoke fundamental freedoms of individuals who were possibly innocent, the world is hardly a place suitable for civilized beings. The standard should be instead the possibility of innocence is required for freedom. How many people in this world have been unfairly punished by people with good intentions doing the best they can with limited resources and time? The same philosophy and approach relates to many other aspects of public policy, and incidentally, the funding of academic resource and questions of computer science. We must focus on positive affirmative outcomes, and processes. Our current justice system says “punish the guilty” and it needs to be saying “protect the innocent”. A system where the old “beyond a reasonable doubt” standard were replaced by a “sure of the possibility of innocence” standard for freedom would be one more fitting for the survival of human beings. The fundamental form of the approach adapted to physics, or mathematics, would help us to continue to innovate and expand out our capabilities, and improve the quality of life for all people. We are an intelligent species, but we need to listen more to anyone who wants to be heard at any time. Astrophysicists should be listening to small town car mechanics. Top universities should require open door policies for the contributions of ideas from outsiders. We require a good to better outline of our futures. If we recognize these goals and accept the responsibilities, the future will be opened to us and we will see we may be able to do things we have only dreamed of in the past, such as discovering new and exciting ways to interact with the universe, share information, and make discoveries. The solution to to the questions of computer science requires these
positive affirmative outcomes, and the same principles must be adapted to physics to continue to innovate and expand out our capabilities.